I wonder if there is a know random variable which is a generalization of the negative binomial random variable. The generalization I search for is with respect to the success probability p -- I want it to vary...

A negative binominal random variable is a sum of geometric random variables with a given success probability p (the same p for all the geometric random variables). I want a type of random variable that is a sum of geometric random variables, but in the case that the geometric random variables are not neccessarily with the same success probability.

An example for such random variable is as follows.

One throws a die (with sides: 1, 2, 3, 4, 5, 6) again and again. A success is when any number shown after throwing the die.
1. The probability for the first success is p1=1. Since one of the 6 numbers will show after throwing the die.
2. After that we erase the number that appeared in the throw! so this side is blank now.
3. Now the second success probability is p2=5/6. Since we alway success unless we got the blank side...
4. after the second success we erase the side that appeared in the thorw.
5. The probability of the third success is p3=4/6. (Two sides are blank...)
6. And so on till the sixth sucess.

The random variable I search for is the number of thorws until, say, the 6'th sucess.